Automated 3D Crack Growth Simulation - Cornell …fac.cfg.cornell.edu/~bruce/papers/ijnme_2000_carter_etal.pdfAutomated 3D Crack Growth Simulation 3 November 18, 2003 process. 3D fracture

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Automated 3D Crack Growth Simulation 1 November 18, 2003

Automated 3D Crack Growth Simulation

B.J. CARTER, P.A. WAWRZYNEK, and A.R. INGRAFFEA

Cornell Fracture Group, 641 Rhodes Hall,

Cornell University, Ithaca, NY, USA 14853

SUMMARY

Automated simulation of arbitrary, non-planar, 3D crack growth in real-life engineered

structures requires two key components: crack representation and crack growth mechanics.

A model environment for representing the evolving 3D crack geometry and for testing

various crack growth mechanics is presented. Reference is made to a specific implementation

of the model, called FRANC3D. Computational geometry and topology are used to represent

the evolution of crack growth in a structure. Current 3D crack growth mechanics are

insufficient; however, the model allows for the implementation of new mechanics. A

specific numerical analysis program is not an intrinsic part of the model; i.e., finite and

boundary elements are both supported. For demonstration purposes, a 3D hypersingular

boundary element code is used for two example simulations. The simulations support the

conclusion that automatic propagation of a 3D crack in a real-life structure is feasible.

Automated simulation lessens the tedious and time-consuming operations that are usually

associated with crack growth analyses. Specifically, modifications to the geometry of the

structure due to crack growth, re-meshing of the modified portion of the structure after crack

growth, and re-application of boundary conditions proceeds without user intervention.

KEYWORDS: three-dimensional, fracture, fatigue simulation


1. INTRODUCTION

Crack growth simulation is the process of modeling crack evolution in a structure through

time or with increasing load. This encompasses all aspects of the modeling process from

initial data preparation to visualization of results, leading to prediction of crack growth and

evaluation of structural integrity. Crack growth simulations are common in many fields of

engineering: microscopic cracks in single crystals1, fatigue cracks in aircraft structures2, and

cracking in large concrete dams3 are just a few examples.

Cracks can affect the integrity and the performance of an engineered structure. Although

cracks can never be eliminated, their detrimental effects at least should be mitigated. To

accomplish this, some form of analysis is needed to determine if, when, and how cracking

evolves. In many cases, back-of-the-envelope calculations are sufficient to decide whether a

crack will grow and whether the structure is safe. In some cases, however, it is necessary to

perform detailed, fully three-dimensional, numerical simulations.

In many engineered structures, observed cracks are non-planar 3D features that nucleate from

areas of high stress concentrations in geometrically complex regions of the structure. Despite

the fact that many observed cracks are truly 3D, they often are idealized as planar 3D or

simply as 2D features. Although this drastically simplifies the simulation, the accuracy of the

predictions based on these idealizations generally has not been well characterized. To

quantify the predictions from these idealized models, it is desirable and necessary to have a

truly-3D crack growth simulator. In addition, where such idealizations are poor or

impossible, detailed 3D simulations can be performed forthwith.

Most commercial stress analysis programs are not suitable for the purpose of simulating 3D

crack growth. Although many commercial programs can perform an accurate stress analysis

of a cracked structure, the subsequent propagation of the crack usually is not a simple


process. 3D fracture simulations have been described in the literature4-11; most of these

concentrate on the numerical analysis, neglecting the issues of representation and automated

propagation. In fact, many of the 3D fracture simulations are really pseudo-3D, as the

modeled crack surface remains planar.

To model an evolving crack efficiently and automatically in a complex 3D structure, one

requires two integral components in a simulator: crack representation and crack growth

mechanics. Representation includes the details of storing the geometry of a cracked body in a

computer and updating the geometric description to reflect crack growth; this includes both

the real geometry and the mathematical representation, i.e., the mesh. Mechanics includes

stress analysis, extraction of relevant crack growth parameters, and determination of the

shape, extent, and direction of crack growth. These two components form the basis for

modeling crack evolution.

A conceptual model of a software framework that allows efficient and automatic simulation

of 3D crack propagation is presented. The representational aspects of crack growth

simulation and the components of the conceptual model are discussed in detail using simple

examples to illustrate the key points. Brief discussions of the mechanics of crack growth are

included when appropriate. A specific implementation, called FRANC3D12-14, is used to

predict stress intensity factors for an evolving 3D crack in two different structures: a non-

planar angled crack in a beam under four-point bending and a planar crack in a rotating

turbine disk.


2. A CONCEPTUAL MODEL OF CRACK GROWTH SIMULATION

Crack growth simulation is an incremental process, where a series of steps is repeated for a

progression of models. Each increment of the simulation relies on previously computed

results and represents one crack configuration. There are four primary collections of data or

databases required for each increment. The first is the representational database, denoted Ri ,

where the subscript identifies the increment number. The representational database contains

a description of the solid model geometry, including the cracks, the boundary conditions, and

the material properties. The representational database is transformed by a discretization

process D to a stress analysis database Ai . The discretization process includes a meshing

function M.

D(Ri ,M(Ri ))� Ai (1)

The analysis database contains a complete, but approximate description of the body, suitable

for input to a solution procedure S, often a finite or boundary element stress analysis

program. The solution procedure S is used to transform the analysis database Ai to an

equilibrium database E i which consists of field variables, such as displacements and stresses,

that define the equilibrium solution. The equilibrium solution should contain field variables

and material state information, and in the context of a crack growth simulation, should also

contain values for stress-intensity factors or other fracture parameters F i for all crack fronts.

S(Ai )� Ei , F i (2)

By means of an update function U, E i in conjunction with Ri is used to create a new

representational model Ri +1 , which includes the crack growth increment. The crack growth

function C, which is part of U, determines the direction and extent of the crack growth

increment.

U E i , Ri ,C(F i )( )� Ri +1 (3)

This sequence of operations, as shown in Figure 1, is repeated until a suitable termination

condition is reached. Results of such a simulation might include one or more of the

following: a final crack geometry, a loading versus crack size history, a crack opening


profile, or a history of the crack-front fracture parameters.

U C

Ri

AiEi Fi,

i

i+1i=i+1

S

MD

Figure 1. Incremental crack growth simulations; i denotes the increment of crack growth.

The conceptual model is incorporated into a software framework called FRANC3D (Figure

2) which encompasses all components of the model except for the stress analysis procedure.

The individual components of the program consist of databases (Ri , Ai , E i , and F i) and

functions (D, M, S, U, and C) that operate on the databases.

3. MODEL REPRESENTATION

Before useful engineering simulations can be performed, the databases Ri , A i , Ei and F i

and the functions D, M, S, U, and C must be defined in terms of computational data

structures and algorithms. This section focuses on the representational database. It primarily

focuses on the aspects of model representation that differ in the context of fracture mechanics

from other applications, and more specifically, it focuses on the implementation of the

representational database within FRANC3D.


Stress/Flow/Thermal Analysis

Model Representation

Discretize

FRANC3D

Analysis Input

UpdateModel

Analysis Output

Figure 2. A conceptual model for crack growth simulation incorporated into a software

framework called FRANC3D.

3.1 Solid Modeling For Crack Growth Simulations

Simulation of crack growth is more complicated than many other applications of

computational mechanics because the geometry and topology of the structure evolve during

the simulation. For this reason, a geometric description of the body that is independent of

any numerical discretization (mesh) is maintained and updated as part of the simulation

process. The geometry database is a vital part of the representational database and contains

an explicit description of the solid model, including the crack. The three most widely used

solid modeling techniques: boundary representation (B-rep), constructive solid geometry

(CSG), and parametric analytical patches (PAP) are equally capable of representing

uncracked geometric models15-17. The B-rep is the most appropriate for representing

evolving 3D cracks, however, for two main reasons.

1. A B-rep modeler stores surfaces and surface geometries explicitly. If explicit

topological adjacency information (as defined in the next section) is available as well,


two topologically distinct surfaces can share a common geometric description. A

crack represents such a configuration. A B-rep modeler uses only the surface

topology and geometry to represent the three-dimensional solid structure. A crack

actually consists of two surfaces that have the same geometric description and both of

these surfaces form part of the solid model boundary.

2. Both 3D solids and dimensionally degenerate forms, such as plates or shells can be

represented equally well with a B-rep modeler. Plates and shells are represented

using a single topological surface to represent each plate/shell surface. This means

that crack growth can be modeled in both solid and thin shell structures using the

same software framework. Note that cracks in shells are mentioned here, but the

primary focus is on 3D solid structures.

3.2 Computational Topology as a Framework for Crack Growth Simulation

The topology of an object is the information about relationships, proximity, and order among

features of the geometry—incomplete geometric information. These are the properties of the

actual geometry that are invariant with respect to geometric transformations (Figure 3). A

topology framework serves as an organizational tool for the data that represents the object

and the algorithms that operate on the data.

Explicit topological information is not essential for crack growth simulation. However, there

are at least four compelling reasons for using topology:

1. Topological information, unlike geometrical information, can be stored exactly, with

no approximations or ambiguity.

2. The theoretical background supporting the concepts of topology and boundary graphs

can be used to develop formal and rigorous procedures for storing and manipulating


these types of data15,16,18.

3. Any topological configuration can represent an infinite number of geometrical

configurations.

4. During crack propagation, the geometry of the structure changes with each crack

increment whereas the topology generally changes less frequently.

x

y

z

topology geometrytied to

vertex

edge

face

Figure 3. Relationship between topology and geometry; a topological entity can have any

number of geometric descriptions.

Prior investigations into the use of data structures for crack propagation simulations19,20

showed that topological databases are a convenient and powerful organizing agent. Topology

allows one to build an efficient system for interactive modeling while hiding the complexities

in manipulating the data. Explicit topological information is used herein as a framework for

Ri and aids in implementation of the functions D and U. Topological entities (such as

vertices, edges, and faces) serve as the principal elements of the database with geometrical

descriptions and other attributes (such as boundary conditions and material properties)

accessed through the topological entities.

Several topological data structures have been proposed for manifold objects. These include:

the winged-edge, the modified winged-edge, the face-edge, the vertex-edge, and the half-edge


data structures16,21,22. A B-rep modeler is capable of representing both solid and thin shell

structures. However, modeling thin shells imposes some constraints on the choice of the

topological data structure. Specifically, non-manifold topologies (Figure 4) are often created;

that is, the topologies cannot exist on a two-manifold representation18. Other features that

introduce non-manifold conditions include internal surfaces, such as bi-material interfaces,

and some crack configurations. A non-manifold data structure is needed to model these types

of structures.

Weiler18 presented an edge-based data structure for storing non-manifold objects, called the

radial-edge, and outlined the corresponding Euler operators. Euler operators are the

fundamental low-level routines for manipulating the data in order to construct or destruct

topological entities. The basic topological entities in the radial-edge data structure are:

vertices, edges, faces, and regions. The edge entity is the primary object through which

topological relationships are maintained and queried (Figure 5). As the name implies, the

edge uses are ordered radially about the edge. Each face has two face uses and each face use

has a corresponding edge use on the given edge. The radial ordering allows for efficient

storage, querying, and manipulation of manifold and non-manifold topology. This data

structure is able to represent model topologies consisting of branching or intersecting cracks

in addition to bi-material interfaces and shell structures – a necessary requirement for a

general purpose fracture simulator.

shell surface

internal void region

wireframeedge

Figure 4. Some examples of non-manifold topologies include internal voids, shell surfaces


attached to solids, and wireframe edges attached to solids.

faceedge

pair of mateface uses

pair of mateedge uses

pair of radialedge uses

vertex

Figure 5. The radial-edge database relies on the radial ordering

of edge-uses about an edge (after 18).

3.3 Topological Representation of a Crack

A crack is defined within Ri by both geometry and topology. A crack generally consists of

multiple faces that are generated as the crack evolves. Crack surfaces are arranged in pairs,

defined as main and mate (Figure 6). Each surface is composed of topological faces, edges

and vertices. Edges and vertices obtain their main/mate classification from their adjacent or

parent faces. The edges and vertices have an additional classification based on their location

on the crack surface (Figure 6). For instance, crack front edges represent the leading edge of

the crack within the solid and open boundary edges represent the intersection of the crack

surface with the model surface. Depending on the presence of the classified edges and

vertices, one can further classify the crack itself.


mainsideface

matesideface

frontedge

open boundary edge

tip vertex

front

front

branchedges

branchvertices

tip vertex

mainmate

Figure 6. The topology of a through and a branching crack.

A single discrete crack in a 3D body is classified as internal, surface, or a discontinuity. An

internal crack has no open boundary edges; the crack front does not intersect the model

boundary surface. A surface crack has at least one open boundary edge and intersects one or

more boundary surfaces. A discontinuity has no crack front edges and completely separates

the body. During propagation, an internal crack can change to a surface crack when the crack

front edges intersect the model boundary surface, and eventually can become a complete

discontinuity (Figure 7).

fully internal crack surface crack

openboundaryedge

discontinuity

Figure 7. An internal crack can change to a surface crack and, eventually, a

complete discontinuity as it propagates.


3.4 Topological Operations for Crack Nucleation and Propagation

The first set of Euler operators that were written for doing crack nucleation and propagation

in FRANC3D are described elsewhere9. These operators are not sufficient for nucleating and

propagating all possible 3D cracks. Therefore, a new set of operators were defined and

implemented. These are the opposite of the glue operators described by Weiler18 and are

called tear operators, consisting of tear vertex, tear edge, and tear face (Figure 8). Tear

vertex creates a mate (V2) vertex with the same geometric location as the original or main

(V1) vertex. Tear edge creates a mate (E2) edge with the same geometric description as the

original or main (E1) edge. Note that the two edges share the same end vertices. Tear face

creates a mate (F2) face with the same geometric description as the original or main (F1) face.

Note that the two faces share the same bounding vertices (and edges).

V1

V2

V3V4

F1tear face

F1V1

V2

V3V4

F2V1

V2

V3

V4

tear vertex

tear edgeV1

V1V2V2E 1

E 1

E 2V1 V2

V1

V1

V2

Figure 8. The individual tear operators.


The process of creating a 3D crack involves at least the tear face operator and can involve the

other two tear operators as well. The tear operators are used to create all types of cracks

including discontinuities and branch-cracks. Cracks in shell and plate structures can be

created using just the tear edge and tear vertex operators. The complementary glue

operators, glue face, glue edge and glue vertex, have also been implemented so that a crack

can be removed from a structure.

The tear operators allow a face, or a set of faces along with their adjacent edges and vertices,

to be torn apart, thereby creating a new crack or propagating an existing crack. This is

accomplished through a sequence of tear operations on the individual topological entities that

comprise the crack:

1) Tear the faces. A tear face operation creates a new mate face (F2 in Figure 8) which

uses the same topological edges and vertices as well as the same geometric description

as the original main face (F1 in Figure 8). A null volume region is created between the

main and mate faces. For example, consider a single isolated face in the interior of a

body. A new internal crack is formed by tearing this face; the bounding edges of the

faces form the crack front.

2) Tear the edges. A tear edge operation is required for any topological edge that is

adjacent to either: two torn faces, or a torn face and the free surface (model boundary).

For example, consider two adjacent faces in the interior of a body (F1 and F2 in Figure

9a). Tearing both faces creates two new null volume regions. In order to make these

two regions contiguous, such that a single crack is formed, the edges (E1 and E2 in

Figure 9b) between the two original main faces must be torn. This produces main (E1

and E2) and mate edges (E3 and E4) on the respective sides of the crack.

3) Tear the vertices. A tear vertex operation is required for any topological vertex that is

common to two torn main edges, excluding vertices that lie on the crack front edge.

For example, crack tip vertices (see Figure 6) are not torn. The vertex (V3) that is


adjacent to the torn edges (E1 and E2 in Figure 9c) must be torn so that the crack

surfaces are completely separated. Vertices that are not adjacent to two torn edges (V1

and V2) are shared by the main and mate faces.

F3F4

F1F2

F1F2 tear faces

F1 and F2

E1 E2

E1 E2

E3 E4

E1 E2

V1 V3

2V3

2

V1 V

V

V1 V4

2V3

2

V1 V

V

tear edgesE1 and E2

tear vertex V3

(a)

(c)

(b)

Figure 9. The tear operators are used in sequence to create a crack.

3.5 Current Solid Modeling Restrictions in FRANC3D

The previous subsections concentrated on the topological description of the model. A few

words about the geometric properties of the topology are required, especially in relation to the

FRANC3D implementation. Although FRANC3D has some solid modeling capabilities, it is

not a true solid modeler and it has some restrictions and limitations that are important in

terms of describing arbitrary crack nucleation and propagation. These are:

1. Geometric surfaces are represented as planar, triangular Bezier, or quadrilateral bi-

cubic B-spline patches. Patches can have three or four bounding edges. A bounding


edge can be formed from multiple topological edges, however.

2. Geometry edges are either straight lines or cubic B-spline curves. In surface parametric

space, all edges are straight lines.

3. Edges can be added to existing faces of the model, but cannot extend beyond the

boundaries of that face (Figure 10a). Multiple edges must be added in such a case.

4. Faces can be created from a set of bounding edges, but adjacent faces must use

common edge geometric descriptions so that faces are topologically congruent. In

other words, topologically adjacent faces cannot have gaps between the common

geometric boundaries.

5. Surfaces can intersect, but they cannot cross through each other (Figure 10b). In that

case, two separate surfaces must be created on opposite sides of the intersected face.

face 1 face 1

face 3

(b) faces are not allowed to cross

face 1 face 2

Not Allowed

edge 1

(a) edges are not allowed to cross face boundaries

NotAllowed

Okay

face 4

face 2

face 2

face 1 face 2

Okay

edge 1 edge 2

Figure 10. (a) Edges cannot cross face boundaries; multiple edges must be added instead.

(b) Faces can intersect, but cannot cross through each other; separate faces must be added.


4. CRACK UPDATE FUNCTION

To represent an evolving crack, a simulator must be capable of both nucleating and

propagating a crack. Crack nucleation consists of placing an initial flaw in the model. Crack

propagation has two components. First, the parameters that govern the crack growth, such as

stress intensity factors, crack growth direction, and amount of extension must be determined.

Second, the topology and geometry must be updated to account for the proper amount of

crack growth. These components are discussed below.

4.1 Techniques for Nucleating Cracks in FRANC3D

Real cracks generally have arbitrary shapes and sizes. Thus, a 3D fracture simulator should

be able to model any crack shape. For many simulations, however, the initial cracks are quite

small with relatively simple geometries. While initial crack geometry is often simple, the

evolutionary geometry can become very complex depending on the stress field and the

geometry of the structure.

An initial assumed or observed crack must be added to the structure geometry. To aid this

process, FRANC3D has a pre-defined library of simple crack shapes (Figure 11) that can be

nucleated automatically. The user can select one of these shapes and then, through a series of

rotations and translations, position the crack in the correct location and orientation. The

translations and rotations, if not defined by observation, could be determined from the

maximum tensile stress direction and the geometry of the structure.


e.g.: a quarter-ellipticalsurface flaw shape

control point

surfaceedge

internal edge(crack front)

31

2

surface edge

Figure 11. FRANC3D library flaw shapes. Each flaw shape is a planar surface and is

defined by the illustrated boundary edges and control points at the ends of the edges.

Although the library cracks have simple planar geometries, adding a crack to an arbitrary 3D

structure can be quite complicated. First, the control points (Figure 11) that define the ends

of the edges must be located in the model and defined as vertices. New vertices are added if

they do not exist in the model at these locations which implies either splitting a geometry

edge or adding a vertex to a face or to a region. Once all the vertices have been found or

created at all of the control points, the edges are added to the model. The edge geometry as

well as the internal or surface classification is specified by the library data. Edges are added

either directly to surfaces or as wireframe edges to regions. Faces are created from the set of

predefined loops of edges as defined by the library data. The tear operators, as described in

Section 3.4, are used to create a crack from the resulting faces, edges, and vertices.

The edges might not exist in the model exactly as specified in the library, however.

Depending on the presence of material interfaces or complexities in the model's surfaces,

multiple topological edges are required rather than the single edge described by the library.

For example, consider a crack surface edge that falls on several non-planar model-bounding

surfaces (Figure 12). The single library edge (1-3 in Figure 11) is broken into segments (1-4,

4-5, and 5-3 in Figure 12). Each edge segment is constrained to lie on the model-bounding


surface and the set of edges must be contiguous across face boundaries. If the single library

edge is broken into several pieces, then the predefined loops of edges that form the crack

faces are invalid. The edges are traversed again to form valid loops and faces before the

crack is finally created.

2

54

3

1

Figure 12. A quarter-elliptical surface flaw from the flaw library is complicated by the model

geometry. The single edge 1-3 shown in Figure 11 is broken into three segments so that it is

consistent with the model surface geometry.

4.2 The Mechanics of Fatigue Crack Growth

To model the evolution of the initial crack to its final state, one needs to decide where and

when the crack will propagate. Assume that a stress analysis for Ri has been performed and

that E i exists. The crack growth model C takes the field values at or near the crack front and

evaluates the potential for further crack growth. The direction of crack growth and the


amount of extension for unique points along the existing crack front defines a set of points in

space that are used to represent the position of the new crack front.

A generally accepted 3D crack growth rule or theory is yet to be developed. The most

common technique for extending a 3D crack is to treat it as a series of 2D plane strain

slices4,23. Although FRANC3D is designed for testing and verifying new theories for 3D

crack growth24, the above approach is used for the two example simulations in this paper.

The crack front is first subdivided based on a user-defined number of points (slices). The

displacement on the crack surface for each subdivision point j along the current crack front i

(see Figure 1) is obtained from the equilibrium state database E i . The displacement is

converted to a fracture parameter F i , more specifically, stress intensity factors

(KI , KII , and KIII ). The direction of extension ( ˆ d ij ) for each point is defined using one of

several existing theories25-27. The relative advance (l ij ) is determined assuming a Paris-law

growth model10,23 with the maximum advance defined by the user.

l ij = Li

K ij

K i max

� � � �

� �

n

(4)

n is the Paris-law exponent, Li is the user-defined maximum extension, K ij is the value of

KI at point j of crack front i, and K i max is the maximum value of KI for crack front i. The

direction ˆ d ij and advance l i

j along with the position ˆ x ij of the points on the crack front are

used to compute the geometry of the new crack front i+1. The model geometry and topology

must be updated to accommodate the crack growth.

4.3 The Representation of Crack Growth

Crack growth invalidates the representational database Ri meaning that updates to the

geometry and the topology are required. The updates are handled by the function U while

local re-discretization and re-meshing are handled by functions D and M, respectively. One


must recognize that crack growth is constrained by both Ri and the crack growth model C.

In other words, the crack surface cannot extend beyond the boundaries of the structure, even

if C predicts such an occurrence. The function U involves adding new vertices, edges, and

faces to the model and then tearing them to produce the new crack surface. Portions of this

function are described next using an internal crack to illustrate the ideas.

Assume that an internal elliptical crack, defined by the shaded area in Figure 13, has been

analyzed and the new crack front points determined as described in Section 4.2. The

propagation of this crack is described next.

Figure 13. An internal elliptical crack is defined by the shaded area.

The set of points (+ symbols) define the predicted new crack front.

4.3.1 Adding The New Crack Front Edges

A polynomial function is fit through the new crack front points. The function can be a single

n-th order polynomial, a continuous piecewise set of n-th order polynomials, or a set of cubic

Hermitian polynomials. The first two are used for surface cracks and the latter is used

exclusively for closed-loop crack fronts such as an internal crack front. A least-squares

routine is used to give the best function in all cases. From the function, a set of fitted points

is determined. The function serves two purposes: 1) it smoothes numerical and geometrical

distortions; and 2) it provides a convenient method for intersecting the crack front with the


model surfaces.

The second step is to determine the region(s) in which the fitted points fall; they might lie

outside the model boundary. A simple ray casting technique28, extending the point-in-

polygon routine17 to 3D, is used for this purpose.

Finally, new edges are added to the model using the fitted points to define the edge geometry.

If the new front points fall in the same region as the original crack, adding the new edges is

straightforward. If the points fall in different regions, then edges will intersect surfaces and

the polynomial functions described above are used to find the intersection points.

Figure 14. New crack front edges added to the model through the fitted points.

The solid squares represent the vertices on the new crack front edges.

4.3.2 Adding new edges to connect old and new crack fronts

Once the new crack front edges have been added, the old crack front must be connected to the

new crack front. For the example presented here, this step is straightforward. Both the old

and new crack fronts are composed of four edges (Figure 14). The vertices on the old crack

front (solid circles) are connected to the vertices on the new crack front (solid squares) by

adding wireframe edges to the region (Figure 15). In general, adding connecting edges can be

more difficult depending on the crack and structure geometry.


Figure 15. Connecting edges between vertices on the old (solid circles) and vertices on the

new (solid squares) crack front are added as wireframe edges.

4.3.3 Adding the loops and faces

After adding the new front and the connecting edges, there exists a set of interconnected

edges consisting of the old front, new front, and connecting edges. The correct loops must be

formed from these edges to create the necessary new crack faces. The loops can have only

three or four bounding edges, a FRANC3D restriction, and they must be valid, i.e., edges

must not cross other edges or loops.

In the example presented here, there are four loops each defined by four bounding edges

(Figure 15). The correct loops can be formed immediately after adding the connecting edges.

In many cases, the formation of valid loops is not this simple, and an algorithm is required

that automatically forms valid loops from an arbitrary set of interconnected edges.

FRANC3D uses an exhaustive search technique along with a number of heuristic checks to

find all the correct loops.

Faces are created automatically for each valid loop of edges. The tear operators defined in

Section 3.4 then are called upon to tear the appropriate faces, edges, and vertices, thereby


propagating the crack.

4.4 Automation

Crack growth as described above consists of two parts: mechanics and representation.

Current crack growth mechanics are based on 2D theories and need to be replaced by an

acceptable 3D theory. The representation, which consists of the topology and geometry and

the associated functions and databases, is completely generic, however; any crack in any 3D

structure can be represented and propagated. In addition, crack propagation is completely

automated in FRANC3D; the user is not required to interact or intervene at any stage after

defining the initial cracked model.

The automated simulations rely on suitable numerical discretization techniques and accurate

analysis codes. FRANC3D supports both finite and boundary elements, but could be

modified to work with other numerical techniques29. Regardless of the numerical technique,

the discretization of the cracked structure must be accomplished such that the evolution of the

fracture is modeled accurately both in terms of numerics and geometry. In addition, the

discretization must be automatic, fast, and robust.

5. DISCRETIZATION AND MESHING

Crack growth simulations cause the geometry and, therefore, the mesh to continually change

as the crack evolves. To limit the time and effort spent on meshing and re-meshing, a

discretization function was designed to minimize the changes due to crack growth. This

function D uses a hierarchy of discretized models and also incorporates an automatic meshing

algorithm for meshing local portions of the model that are affected by crack growth.


5.1 A Hierarchy of Models and Constraints

Thus far, only two independent representations of a structure, the geometry and mesh models

were mentioned. Actually, a hierarchy of five models is used in FRANC3D. Within the

model hierarchy is a strong notion of constraint. That is, entities at any level of the hierarchy

are constrained by those in the levels above it.

The geometry (Figure 16a) and mesh (Figure 16e) descriptions represent the highest and

lowest levels in the five-level hierarchy, respectively. The second highest level is a volume

decomposition level (Figure 16b) where regions of the geometric model can be divided into

subregions. This level is useful for defining regularly shaped regions for the volume meshing

algorithm. The third highest level is a surface decomposition level (Figure 16c). This level

is useful when decomposing an irregularly shaped surface into subsurfaces for surface

meshing purposes. The next level is the edge subdivision level (Figure 16d). This level has

two purposes. The spline space-curves of the three higher levels are approximated by

straight-line segments, and the subdivision edge size defines a metric for the local mesh

density. The lowest level in the hierarchy is the mesh, whether it’s a surface or volume mesh,

or a combination. All levels are constrained by the levels above, meaning that the lowest

mesh level has element edges that correspond to the edge subdivision level edges, element

faces cannot span surface subdivision boundaries, element volumes cannot span region

subdivision boundaries, and the entire mesh is constrained by the original geometry.

There are two primary purposes for such a hierarchy. One is to provide constraints and

inheritance for interactive modeling, and the other is to aid in minimizing changes to the

model during crack evolution.

The hierarchy allows the mesh level model to inherit the geometry and the simulation

attributes. Simulation attributes, which consist of boundary conditions and material


properties, are part of the original geometry model. The stress analysis procedure S requires

this data, which means that the mesh must inherit the data from the geometry.

(b)(a)Reg 1

Reg 2

Reg 3

(c)

S5

S4

S3

S2S1

S6

(d) (e)

Figure 16. Constrained model hierarchy, (a) geometry, (b) volume decomposition,

(c) surface decomposition, (d) edge subdivision, and (e) mesh.

During crack evolution, it is desirable that changes to the model between crack growth

increments be kept to a minimum. When a portion of a model is modified, a certain amount

of related information becomes obsolete. To make the overall simulation process as efficient

as possible, the amount of lost information must be minimized. For example, in the case of

crack propagation, some modifications to the geometry are made in the region near the crack

front which invalidates the mesh in this region. However, portions of the model remote from

the crack should not be affected. The total simulation time can be reduced significantly if

only a small portion of the model requires remeshing after each crack increment. The five

levels of model representation provide a convenient hierarchical framework for enforcing this

concept. For instance, a crack growing only in Reg 3 of Figure 16 will not affect the other


two regions, meaning that these two regions do not have to be re-discretized after every step

of propagation.

5.2 Meshing During Crack Growth Simulations

The purpose of the discretization function D is to transform a geometry database to an

analysis database that meets the input requirements of a particular stress analysis procedure.

The major portion of this task is the creation of a surface or volume mesh. An automatic

mesh generation capability can be employed. These are well described in the literature30-35.

There are two aspects of mesh generation that are important in the context of crack growth

simulation that may be less important in other applications. The first arises due to the

geometric coincidence of crack faces. A meshing algorithm used to mesh a surface or

volume containing crack faces, where the crack faces are represented by two distinct surfaces,

cannot rely on geometrical checks exclusively while generating elements. This is because

nodal points on opposing crack faces are distinct, but share a common location. It cannot be

determined from geometrical checks alone if a candidate node is on the proper side of a

crack. To properly mesh such regions, algorithms must resort to topological information to

select the proper node.

The second aspect of mesh generation is important when trying to minimize the changes to

the model during crack evolution. Often only a small portion of the body near the crack front

needs remeshing. However, the new mesh must conform to the remaining unchanged

portions of the mesh, which imposes additional constraints on the meshing algorithm. The

surface meshing algorithm implemented in FRANC3D was described by Potyondy et al. 34 .

A tetrahedral volume meshing algorithm using similar ideas has been implemented as well36.

FRANC3D maintains a consistent geometric representation of the model at each step of


propagation. During fracture propagation, the previous crack surface geometry remains the

same; new fracture surface is simply added to the model to represent the crack growth.

Therefore, the mesh that is attached to the existing geometric crack surfaces is unaffected by

fracture growth because the existing geometry does not change. Actually, the mesh is

removed from the geometric crack surface during propagation, but an identical mesh can be

regenerated on that surface. A new mesh is attached to the new crack surface. The re-

discretization process has been automated completely so that crack growth simulations can

proceed from an initial cracked model without any user intervention.

6. ANALYSIS DATABASES

The stress analysis function S can be any numerical analysis procedure that takes in the

analysis database Ai and produces the equilibrium state information E i and the required

fracture parameters F i . The main requirement of the numerical method is the accurate

calculation of the displacements and stresses near the crack front. Both finite and boundary

element procedures have been developed for this purpose. For the example simulations

below, BES, a 3D linear elastic boundary element code37 is used, which means that only

surface meshing is required.

7. CRACK GROWTH SIMULATIONS

The ability of the above model to simulate crack propagation in 3D is best shown by practical

example. The purpose of this paper is to show that automated simulation of the complete 3D

crack growth evolution in a real-life engineered structure is possible with the appropriate

tools. Accordingly, two simulations are presented: one shows a simple geometry with a non-

planar crack surface, and the other shows a complex geometric structure with a planar crack

surface. In both cases, the crack transitions from a simple half-penny or quarter-penny


shaped surface flaw to a part-through crack. For both examples, the stress intensity factor

history and fatigue life are presented and compared to experimental results after a number of

crack growth increments have been analyzed.

7.1 Beam in Bending with a Non-Planar Crack

The first example is an aluminum beam tested in four-point bending with an initial angled

and offset half-penny shaped crack on the bottom surface. The test configuration and results

are described by Riddell38 so the automated crack growth simulation results will only be

summarized here. The original cracked model (Figure 17) was re-analyzed using the

automated features of FRANC3D; 15 steps of crack growth were simulated.

0.3 m

side view0.05 m

bottom view45° angle0.038 m

αααα ββββ

expandedsection view

initial crackcornercorner

crack

Figure 17. Sketch of the four-point-bend beam with the

initial 45°-angled half-penny shaped crack38.


Upon loading, the crack immediately begins turning so that it becomes perpendicular to the

direction of maximum tension, Figure 18. Between crack growth steps 6 and 7, the half-

penny shaped crack transitions around the αααα-corner of the beam to become a corner crack.

This happens at one side of the beam first because of the initial offset from the beam center.

In this case, the numerical transition is simple because the crack front of the previous step

happened to intersect exactly at the corner of the beam.

Figure 18. The angled crack turns to align itself normal to the principal tensile stress field

and eventually transitions from a half-penny surface crack to a corner crack.

Between steps 13 and 14, the crack transitions into a part-through crack as the crack

propagates around the ββββ-corner of the beam (Figure 19). In this case, two new faces are

required to represent the new crack surface in FRANC3D. An extra edge connecting the new

crack tip to the old crack front is added automatically so that two faces, each composed of

four edges, are created.

Further steps of propagation allow the crack front to become essentially straight as the crack


approaches the upper surface of the beam, becoming a simple part-through type of crack. The

automated simulation ended at propagation step 15.

Figure 19. Final predicted crack shape. The crack transitions around

the second corner using two geometric surfaces.

Riddell38 has compared the FRANC3D/BES simulation results with the experimental results

in detail. The simulated final crack shape compares well with the observed final shape

(Figure 20), as do the surface trace of crack tip locations and the predicted fatigue lives38.

7.2 Rotating Turbine Disk with Planar Crack

The second example is a quarter model of a spinning turbine disk. Such a model was

considered by Mahorter et al.39 who examined the fatigue life for various crack locations in

two sets of turbine disks. A similar turbine disk model (Figure 21) was created and analyzed

here. The disk is subjected to a cyclic rotational velocity between 500 and 11,300 revolutions

per minute, and the assumed material properties are given in Table 1.


Figure 20. Final observed crack shape. The shiny fracture surface is the fatigue crack;

the duller surface is a fast fracture.


Figure 21. B-rep solid quarter-symmetry model of the turbine disk. Table 1. Disk Material Properties

elastic modulus 1.7e4 ksi (117 GPa) Poisson’s ratio 0.33 mass density 4.169e-4 lbf s2/in4 (0.161 kg/m3) ultimate tensile stress 145 ksi (1000 MPa) uniaxial yield stress 135 ksi (931 MPa) part through toughness 65 ksi in.5 (71.4 MPa m.5) plane strain toughness 50 ksi in.5 (55.0 MPa m.5) plane stress/strain parameter Ak 1.0 plane stress/strain parameter Bk 0.5 da/dN parameter C 3.11e-10 in/cycle (1.77e-14 m/cycle) da/dN parameter n 3.667 da/dN parameter p 0.25 da/dN parameter q 0.75 threshold deltaK (R=0) 3.5 ksi in.5 (3.85 MPa m.5) closure R 0.7 constraint parameter alpha 2.5 stress ratio Smax/Sflow 0.3

An initial crack was assumed to start from the bolthole indicated in Figure 21. An initial

quarter-penny shaped corner crack of 0.76 mm radius was created using the FRANC3D

library flaw (Figure 22). The crack is aligned with the radial direction of the disk. The initial

model was meshed and boundary conditions were applied using the maximum rotational

velocity and constraints on the symmetry surfaces. The model was prepared to run

automatically for 10 crack growth analyses.

The corner crack propagates in the original radial plane growing both through the disk

thickness and radially. Between step 6 and 7, the corner crack transitions into a part-through

crack as the crack extends completely through the thickness of the disk (Figure 23). The

crack continues to propagate radially as a part-through crack until the analyses stop.


Figure 22. Initial quarter penny-shaped corner crack located at the base of the bolthole.

Figure 23. The crack extends through the disk thickness and becomes a part-through crack as

it continues to propagate radially into the disk.


Figure 24 shows the Mode I SIF history for a single trace along the crack surface. The SIF

varies along the crack front; thus, the SIF history and the predicted fatigue life will vary

depending on the chosen path. Using the Forman-Newman-deKoning40,41 fatigue crack

growth model and the material parameters from Table 1, the predicted fatigue life for the disk

starting from an initial corner crack on the bolthole-disk surface with radius 0.76 mm is about

1,300 cycles. Mahorter et al.39 report that the low cycle fatigue (LCF) life of the disk ends

when the crack in the bolthole reaches 1/32 inch (0.79 mm); this corresponds to the B.1 life

which is expressed as a 1/1000 probability of failure. The analyses presented here provide an

estimate of remaining life of the disk once the LCF life is reached. Due to the uncertainty in

measuring LCF life and in detecting 1/32 inch flaw sizes, any significant amount of disk life

after LCF serves to increase the confidence in using these turbine disks. Additional

simulations could be performed to estimate the LCF life as well, but this is beyond the scope

and purpose of this paper.

0 1 2 3 4 5 620

30

40

50

60

70

80

crack length (mm)

Mod

e I S

IF (M

pa m

)

Figure 24. The Mode I SIF history for a path along the crack surface.


.

0 200 400 600 800 1000 1200 14000

1

2

3

4

5

6

0 200 400 600 800 1000 1200 14000

1

2

3

4

5

6

number of cycles

crac

k le

ngth

(mm

)

Figure 25. The predicted fatigue life of the turbine disk starting from

an initial corner crack in the bolthole of radius 0.76 mm.

8. SUMMARY AND CONCLUSIONS

The software framework, specifically FRANC3D, is designed for modeling arbitrary crack

growth in three-dimensional solid and shell structures. The use of a boundary representation

to describe the structure, a topological database to store and manipulate the data, and the

constrained hierarchy of models to simplify the discretization serve to make the abstract

model for crack growth simulations a versatile tool for performing crack growth simulations.

Crack propagation is controlled by the structure geometry and the loading. Automatic

propagation of an arbitrary 3D crack in an arbitrary 3D structure is possible through the use

of both topological and geometrical operations. Although it is straight-forward for a user to

visualize the new crack faces, edges, and vertices that must be created to simulate crack

propagation, it is much more difficult to automate this procedure to work for all cases of

crack and structure geometry. By combining various aspects of computer graphics and


intelligent computational algorithms with some heuristic and logical checks, one can create a

system that is capable of handling very complex fracture propagation in a generic sense. Of

course, one needs a sophisticated topological database, such as the radial-edge database,

along with all the necessary Euler operators to represent, query, and efficiently manipulate

the model data.

All of the apparently tedious and time-consuming tasks: propagation, re-meshing, and re-

applying boundary conditions can be completely automated. The entire evolution of crack

growth can be simulated providing histories of stress intensity factors and predicted crack

shapes from which one can compute fatigue life. The software framework easily allows other

physical models for 3D crack growth to be implemented and tested, and new theories can

readily be compared to experimental or field studies by modeling the true structure and crack

geometry rather than an overly simplified or idealized model.

ACKNOWLEDGMENTS

Funding for the creation of FRANC3D has been provided in part by NASA (NAG-1-1184),

NSF (PYI 8351914), Boeing Commercial Airplane Group, Alcoa Foundation, Northrop

Grumman, and Schlumberger. The authors acknowledge the significant contributions of the

present and former members of the Cornell Fracture Group (CFG) in developing this

software and the concepts behind it. The software and documentation are freely available

from the CFG at http://www.cfg.cornell.edu.

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